Chapter 10 Comparisons Involving Means Part A

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Chapter 10 Comparisons Involving Means Part A. Estimation of the Difference between the Means of Two Populations: Independent Samples Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples.

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Chapter 10 Comparisons Involving MeansPart A
• Estimation of the Difference between the Means of Two Populations: Independent Samples
• Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples
Estimation of the Difference Between the Means of Two Populations: Independent Samples
• Point Estimator of the Difference between the Means of Two Populations
• Sampling Distribution of
• Interval Estimate of Large-Sample Case
• Interval Estimate of Small-Sample Case

Let equal the mean of sample 1 and equal the

mean of sample 2.

• The point estimator of the difference between the
• means of the populations 1 and 2 is .

Point Estimator of the Difference Betweenthe Means of Two Populations

• Let 1 equal the mean of population 1 and 2 equal

the mean of population 2.

• The difference between the two population means is

1 - 2.

• To estimate 1 - 2, we will select a simple random

sample of size n1 from population 1 and a simple

random sample of size n2 from population 2.

Sampling Distribution of

• Expected Value
• Standard Deviation

where:1 = standard deviation of population 1

2 = standard deviation of population 2

n1 = sample size from population 1

n2 = sample size from population 2

• Interval Estimate with 1 and 2 Known

where:

1 -  is the confidence coefficient

• Interval Estimate with 1 and 2 Unknown

where:

• Example: Par, Inc.

Par, Inc. is a manufacturer of golf

equipment and has developed

a new golf ball that has been

designed to provide “extra

distance.” In a test of driving

distance using a mechanical

driving device, a sample of

Par golf balls was compared with a sample of golf balls

made by Rap, Ltd., a competitor.

The sample statistics appear on the next slide.

• Example: Par, Inc.

Sample #1

Par, Inc.

Sample #2

Rap, Ltd.

Sample Size

120 balls 80 balls

Sample Mean

235 yards 218 yards

Sample Std. Dev.

15 yards 20 yards

Simple random sample

of n1 Par golf balls

x1 = sample mean distance

for sample of Par golf ball

Simple random sample

of n2 Rap golf balls

x2 = sample mean distance

for sample of Rap golf ball

x1 - x2 = Point Estimate of m1 –m2

Point Estimator of the Difference Betweenthe Means of Two Populations

Population 1

Par, Inc. Golf Balls

m1 = mean driving

distance of Par

golf balls

Population 2

Rap, Ltd. Golf Balls

m2 = mean driving

distance of Rap

golf balls

m1 –m2= difference between

the mean distances

Point Estimate of the DifferenceBetween Two Population Means

Point estimate of 1-2 =

= 235 - 218

= 17 yards

where:

1 = mean distance for the population

of Par, Inc. golf balls

2 = mean distance for the population

of Rap, Ltd. golf balls

95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown

Substituting the sample standard deviations for the population standard deviation:

17 + 5.14 or 11.86 yards to 22.14 yards

We are 95% confident that the difference between

the mean driving distances of Par, Inc. balls and Rap,

Ltd. balls is 11.86 to 22.14 yards.

Using Excel to Develop anInterval Estimate of m1 – m2: Large-Sample Case

• Formula Worksheet

Note: Rows 16-121 are not shown.

Using Excel to Develop anInterval Estimate of m1 – m2: Large-Sample Case

• Value Worksheet

Note: Rows 16-121 are not shown.

Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30)
• Interval Estimate with  2 Unknown

where:

Difference Between Two Population Means:

Small Sample Case

• Example: Specific Motors
• Specific Motors of Detroit

has developed a new automobile

known as the M car. 12 M cars

and 8 J cars (from Japan) were road

tested to compare miles-per-gallon (mpg)

performance. The sample statistics are shown on the

next slide.

Difference Between Two Population Means:

Small Sample Case

• Example: Specific Motors

Sample #1

M Cars

Sample #2

J Cars

12 cars 8 cars

Sample Size

29.8 mpg 27.3 mpg

Sample Mean

2.56 mpg 1.81 mpg

Sample Std. Dev.

Point Estimate of the DifferenceBetween Two Population Means

Point estimate of 1-2 =

= 29.8 - 27.3

= 2.5 mpg

where:

1 = mean miles-per-gallon for the

population of M cars

2 = mean miles-per-gallon for the

population of J cars

95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case

We will make the following assumptions:

• The miles per gallon rating is normally
• distributed for both the M car and the J car.
• The variance in the miles per gallon rating
• is the same for both the M car and the J car.

95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case

• We will use a weighted average of the two sample
• variances as the pooled estimator of  2.
95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case

2.5 + 2.2 or .3 to 4.7 miles per gallon

• Using the t distribution with n1 + n2 - 2 = 18 degrees
• of freedom, the appropriate t value is t.025 = 2.101.

We are 95% confident that the difference between

the mean mpg ratings of the two car types is .3 to 4.7

mpg (with the M car having the higher mpg).

• Formula Worksheet
• Value Worksheet
Hypotheses

Test Statistic

Hypothesis Tests About the Difference between the Means of Two Populations: Independent Samples

Large-Sample

Small-Sample

Between the Means of Two Populations:

Independent Samples, Large-Sample Case

• Example: Par, Inc.

Recall that Par, Inc. has

developed a new golf ball that

was designed to provide “extra

distance.” A sample of Par golf

balls was compared with a sample of golf balls made

by Rap, Ltd., a competitor.

The sample statistics appear on the next slide.

Between the Means of Two Populations:

Independent Samples, Large-Sample Case

• Example: Par, Inc.

Can we conclude, using a = .01,

that the mean driving distance of

Par, Inc. golf balls is greater than

the mean driving distance of

Rap, Ltd. golf balls?

Sample #1

Par, Inc.

Sample #2

Rap, Ltd.

Sample Size

120 balls 80 balls

Sample Mean

235 yards 218 yards

Sample Std. Dev.

15 yards 20 yards

Between the Means of Two Populations:

Independent Samples, Large-Sample Case

• Using the Test Statistic

1. Determine the hypotheses.

H0: 1 - 2< 0

Ha: 1 - 2 > 0

where:

1 = mean distance for the population

of Par, Inc. golf balls

2 = mean distance for the population

of Rap, Ltd. golf balls

Between the Means of Two Populations:

Independent Samples, Large-Sample Case

• Using the Test Statistic

a = .01

2. Specify the level of significance.

3. Select the test statistic.

4. State the rejection rule.

Reject H0 if z > 2.33

Between the Means of Two Populations:

Independent Samples, Large-Sample Case

• Using the Test Statistic

5. Compute the value of the test statistic.

Between the Means of Two Populations:

Independent Samples, Large-Sample Case

• Using the Test Statistic

6. Determine whether to reject H0.

z = 6.49 > z.01 = 2.33, so we reject H0.

At the .01 level of significance, the sample evidence

indicates the mean driving distance of Par, Inc. golf

balls is greater than the mean driving distance of Rap,

Ltd. golf balls.

… continued

Using Excel to Conduct aHypothesis Test about m1 – m2: Large Sample Case

• Excel’s “z-Test: Two Sample for Means” Tool

Step 2Choose the Data Analysis option

Step 3 Choose z-Test: Two Sample for Means

from the list of Analysis Tools

… continued

Using Excel to Conduct aHypothesis Test about m1 – m2: Large Sample Case

• Excel’s “z-Test: Two Sample for Means” Tool

Step 4When the z-Test: Two Sample for Means

dialog box appears:

Enter A1:A121 in the Variable 1 Range box

Enter B1:B81 in the Variable 2 Range box

Type 0 in the Hypothesized Mean

Difference box

Type 225 in the Variable 1 Variance

(known) box

Type 400 in the Variable 2 Variance

(known) box

Using Excel to Conduct aHypothesis Test about m1 – m2: Large Sample Case

• Excel’s “z-Test: Two Sample for Means” Tool

Step 4 (continued)

Select Labels

Type .01 in the Alpha box

Select Output Range

Enter D4 in the Output Range box

(Any upper left-hand corner cell indicating

where the output is to begin may be entered)

Click OK

Using Excel to Conduct aHypothesis Test about m1 – m2: Large Sample Case

• Value Worksheet

Note: Rows 16-121 are not shown.

Using Excel to Conduct aHypothesis Test about m1 – m2: Large Sample Case

• Using the p -Value

4. Compute the value of the test statistic.

The Excel worksheet states z = 6.48

5. Compute the p–value.

The Excel worksheet states p-value = 4.501E-11

6. Determine whether to reject H0.

Because p–value < a = .01, we reject H0.